Physics · Nuclear Physics · A2 Level

The Nuclear Atom & Rutherford Scattering

In 1909, two of Ernest Rutherford's students fired alpha particles at gold foil thinner than a fingernail, expecting them to sail straight through a uniform, evenly-spread mass of positive charge. Almost all of them did. But a tiny fraction bounced almost straight back — a result so unexpected that Rutherford compared it to firing a shell at tissue paper and having it rebound. That single experiment demolished the accepted model of the atom overnight.

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The plum pudding model it replaced

Before 1909, the accepted picture of the atom was J. J. Thomson's plum pudding model: a sphere of positive charge, spread fairly uniformly across the atom's volume, with negatively-charged electrons studded through it like plums in a pudding, held by electrostatic attraction to the surrounding positive "dough". Crucially, this model has no concentrated mass or charge anywhere — everything is smeared out evenly, and the atom is solid and continuous throughout.

If this picture were correct, fast, heavy, positively-charged alpha particles fired at a thin sheet of atoms should experience only weak, gradual electrostatic forces as they pass through — the charge they interact with at any instant is spread thin, never concentrated. The expectation was that every alpha particle would pass through with at most a small, gentle deflection.

The Geiger–Marsden alpha-scattering experiment

Working under Rutherford's direction, Hans Geiger and Ernest Marsden directed a narrow, collimated beam of alpha particles at an extremely thin sheet of gold foil — gold was chosen because it can be hammered into a foil only a few hundred atoms thick, thin enough that most alpha particles interact with at most one nucleus on the way through. A rotatable detector — a zinc sulfide screen that flashes (scintillates) whenever an alpha particle strikes it — could be positioned at any angle around the foil, including beyond 90°, to record how many particles emerged in each direction.

What was observed

The overwhelming majority of alpha particles passed straight through the foil, or were deflected through only very small angles.
A small fraction — roughly 1 in 8000 for the original experiment — were deflected through large angles, some greater than 90°.
An extremely small fraction bounced back almost the way they had come, at angles close to 180° (backscattering).

An animated diagram showing nine alpha particles fired from the left at a thin vertical strip of gold foil with a small solid black circle at its centre representing one gold nucleus. Six particles, entering well above or below the nucleus, travel straight through the foil undeflected. Two particles, entering close to the nucleus, bend through a moderate angle as they cross the foil and continue onward at a new angle. One particle, entering almost exactly along the centre line, approaches the nucleus, curves sharply around just short of touching it, and travels back out the way it came, passing close beside its own incoming path. — Rutherford scattering: most alpha particles pass straight through the foil undeflected; a small number are deflected by a nearby nucleus; a very small number, on a near head-on path, are scattered back almost the way they came.

What the observations imply

Each of the three observations rules out the plum pudding model and points toward a different atomic structure.

Most alpha particles pass straight through — the atom is mostly empty space. If positive charge and mass were spread evenly throughout, every alpha particle would feel a comparable force throughout its passage and none would travel through undeflected.
A small number are deflected through large angles — this can only happen if an alpha particle occasionally passes very close to a concentration of positive charge intense enough to repel it strongly. This implies the atom's positive charge (and essentially all its mass, since alpha particles are far lighter than a whole atom but easily deflected) is concentrated in a tiny volume.
A very small number bounce almost straight back — for a fast alpha particle to reverse direction almost entirely, it must experience an enormous repulsive force, momentarily overcoming all its kinetic energy. This could only happen on a rare near head-on approach to a positive charge comparable to or greater than the alpha particle's own mass, concentrated in a region small enough for the force to become that large.

The nuclear model of the atom

Putting the three observations together gives the nuclear model: almost all of an atom's mass and all of its positive charge are concentrated in a tiny, dense nucleus at the centre, with the rest of the atom — many thousands of times larger in radius — essentially empty space through which much lighter, negatively-charged electrons are arranged.

Estimating the size of the nucleus

The closest a head-on alpha particle gets to the nucleus before its kinetic energy is entirely converted into electrostatic potential energy — the distance of closest approach — gives an upper estimate for the nuclear radius. At that instant the alpha particle is momentarily at rest, so all its initial kinetic energy has transformed into electric potential energy of the alpha–nucleus system.

DerivationDistance of closest approach, from KE = electric PE

An alpha particle of charge $q_\alpha = +2e$ approaches a nucleus of charge $Q = +Ze$ head-on, where $Z$ is the nucleus's proton number.

The alpha particle is a helium nucleus, so it carries two elementary charges.

Far from the nucleus, the alpha particle has kinetic energy $E_k = \tfrac{1}{2}mv^2$, and electric potential energy is negligible.

At the distance of closest approach $r_0$, the alpha particle is momentarily at rest, so all of $E_k$ has converted into electric potential energy: $E_k = \dfrac{Q q_\alpha}{4\pi\varepsilon_0 r_0} = \dfrac{Ze \times 2e}{4\pi\varepsilon_0 r_0}$.

Electric potential energy between two point charges is $E_p = \dfrac{Q_1 Q_2}{4\pi\varepsilon_0 r}$.

Rearranging for $r_0$: $r_0 = \dfrac{2Ze^2}{4\pi\varepsilon_0 E_k}$.

For gold ($Z=79$) and typical alpha-particle energies of a few MeV, this gives $r_0$ of the order $10^{-14}\,\text{m}$ — thousands of times smaller than the accepted radius of the atom itself, around $10^{-10}\,\text{m}$.

$r_0$ is only an upper bound on the nuclear radius: it tells you the alpha particle got no closer than this, so the nucleus can be no bigger than $r_0$. Higher-energy alpha particles, which penetrate closer before being stopped, give a tighter (smaller) estimate, until at very high energies the alpha particle starts to feel the strong nuclear force and the simple electrostatic argument breaks down.

▪ derived

Why this is an upper bound, not an exact measurement

This calculation assumes the alpha particle only ever feels electrostatic (Coulomb) repulsion from the nucleus — a good approximation as long as it never gets close enough to feel the short-range strong nuclear force. If it did, the result would depart from the smooth $1/r$ curve predicted by Coulomb's law, and this departure — first seen with higher-energy alpha particles — was itself evidence for the existence and range of the strong nuclear force.

Why the plum pudding model couldn't survive this

It wasn't that alpha particles were deflected at all — a smeared-out positive charge could, in principle, produce very slight deflections through gentle, cumulative nudges as the alpha particle passed through many atoms. What was fatal was the existence of any large-angle scattering events, however rare. A diffuse charge distribution cannot produce the enormous, localised force needed to reverse an alpha particle's direction in a single close encounter. Only a model with a small, dense, highly-charged core can explain even one such event, and Geiger and Marsden observed enough of them, at a high enough rate, to make the conclusion unavoidable.

Key takeaways

The plum pudding model pictured the atom as a uniform sphere of positive charge with embedded electrons — no concentrated mass or charge anywhere.
In the Geiger–Marsden experiment, alpha particles were fired at thin gold foil: most passed straight through, a small fraction deflected through large angles, and a very small fraction backscattered almost the way they came.
These observations imply the atom is mostly empty space, with almost all its mass and positive charge concentrated in a tiny, dense central nucleus — the nuclear model of the atom.
The distance of closest approach, found from equating an alpha particle's kinetic energy to the electric potential energy at the point it momentarily stops, $E_k = \dfrac{2Ze^2}{4\pi\varepsilon_0 r_0}$, gives an upper-bound estimate for the nuclear radius — of order $10^{-14}\,\text{m}$, thousands of times smaller than the atom itself.
This estimate assumes purely electrostatic repulsion; departures from it at very close approach were early evidence for the strong nuclear force.

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